Hyperbolic traveling waves driven by growth

Emeric Bouin∗, Vincent Calvez†‡, Gregoire Nadin§

July 8, 2013

Abstract

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPPequation. This model accounts for particles that move at maximal speed ε−1 (ε > 0), andproliferate according to a reaction term of monostable type. We study the existence andstability of traveling fronts. We exhibit a transition depending on the parameter ε: for smallε the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However,for large ε the traveling front with minimal speed is discontinuous and travels at the maximalspeed ε−1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces.We also prove local nonlinear stability of the traveling front with minimal speed when ε issmaller than the transition parameter.

Key-words: traveling waves; Fisher-KPP equation; telegraph equation; nonlinear stability.AMS classification: 35B35; 35B40; 35L70; 35Q92

1 Introduction

We consider the problem of traveling fronts driven by growth (e.g. cell division) together with celldispersal, where the motion process is given by a hyperbolic equation. This is motivated by theoccurence of traveling pulses in populations of bacteria swimming inside a narrow channel [1, 44].It has been demonstrated that kinetic models are well adapted to this problem [45]. We will focuson the following model introduced by Dunbar and Othmer [13], Hadeler [27], Holmes [29], Mendezand co-authors [33, 34, 22, 39], and Fedotov [16, 17, 18] (see also the recent book [35]),

ε2∂ttρε(t, x) +(1− ε2F ′(ρε(t, x))

)∂tρε(t, x)− ∂xxρε(t, x) = F (ρε(t, x)) , t > 0 , x ∈ R . (1.1)

The cell density is denoted by ρε(t, x). The parameter ε > 0 is a scaling factor. It accounts for theratio between the mean free path of cells and the space scale. The growth function F is subjectto the following assumptions (the so-called monostable nonlinearity){

F ∈ C3([0, 1]) , F is uniformly strictly concave : inf [0,1](−F ′′) =: α > 0 ,

F (0) = F (1) = 0 , F (ρ) > 0 if ρ ∈ (0, 1) .(1.2)

For the sake of clarity we will sometimes take as an example the logistic growth function F (ρ) =ρ(1− ρ).

Equation (1.1) is equivalent to the hyperbolic system{∂tρε + ε−1∂x (jε) = F (ρε)ε∂tjε + ∂xρε = −ε−1jε .

(1.3)

∗Ecole Normale Superieure de Lyon, UMR CNRS 5669 ’UMPA’, 46 allee d’Italie, F-69364 Lyon cedex 07, France.E-mail: emeric.bouin@ens-lyon.fr†Corresponding author‡Ecole Normale Superieure de Lyon, UMR CNRS 5669 ’UMPA’, and INRIA project NUMED, 46 allee d’Italie,

F-69364 Lyon cedex 07, France. E-mail: vincent.calvez@ens-lyon.fr§Universite Pierre et Marie Curie-Paris 6, UMR CNRS 7598 ’LJLL’, BC187, 4 place de Jussieu, F-

75252 Paris cedex 05, France. E-mail: nadin@ann.jussieu.fr

1

The expression of jε can be computed explicitly in terms of ρε as follows,

jε(t, x) = −1

ε

∫ t

0

∂xρε(s, x) exp

(s− tε2

)ds+ jε(0, x) , (1.4)

but this expression will not be directly used afterwards. We will successively use the formulation(1.1) or the equivalent formulation (1.3).

Since the pioneering work by Fisher [21] and Kolmogorov-Petrovskii-Piskunov [31], dispersionof biological species has been usually modelled by mean of reaction-diffusion equations. Themain drawback of these models is that they allow infinite speed of propagation. This is clearlyirrelevant for biological species. Several modifications have been proposed to circumvent this issue.It has been proposed to replace the linear diffusion by a nonlinear diffusion of porous-medium type[46, 36, 41]. This is known to yield propagation of the support at finite speed [37, 38]. The density-dependent diffusion coefficient stems for a pressure effect among individuals which influences thespeed of diffusion. Pressure is very low when the population is sparse, whereas it has a strongeffect when the population is highly densified. Recently, this approach has been developped forthe invasion of glioma cells in the brain [7]. Alternatively, some authors have proposed to imposea limiting flux for which the nonlinearity involves the gradient of the concentration [3, 8, 4].

The diffusion approximation is generally acceptable in ecological problems where space andtime scales are large enough. However, kinetic equations have emerged recently to model self-organization in bacterial population at smaller scales [2, 40, 14, 32, 42, 44, 45]. These models arebased on velocity-jump processes. It is now standard to perform a drift-diffusion limit to recoverclassical reaction-diffusion equations [28, 10, 14, 30]. However it is claimed in [45] that the diffusionapproximation is not suitable, and the full kinetic equation has to be handled with. Equation (1.1)can be reformulated as a kinetic equation with two velocities only v = ±ε−1 (see (2.1) below).This provides a clear biological interpretation of equation (1.1) as a simple model for bacteriacolonies where bacteria reproduce themselves, and move following a run-and-tumble process. Wealso emphasize that model (1.1) arises in the biological issue of species range expansion [29, 39],and in particular the human Neolithic Transition [22].

Hyperbolic models coupled with growth have already been studied in [13, 27, 24, 11]. In [27] itis required that the nonlinear function in front of the time first derivative ∂tρε is positive (namelyhere, 1− ε2F ′(ρ) > 0). Indeed, this enables to perform a suitable change of variables in order toreduce to the classical Fisher-KPP problem. In our context this is equivalent to ε2F ′(0) < 1 sinceF is concave. In [24] this nonlinear contribution is replaced by 1: the authors study the followingequation (damped hyperbolic Fisher-KPP equation),

ε2∂ttρε(t, x) + ∂tρε(t, x)− ∂xxρε(t, x) = F (ρε(t, x)) .

We also refer to [11] where the authors analyse a kinetic model more general than (1.1). Theydevelop a perturbative approach, close to the diffusive regime ε� 1.

It is worth recalling some basic results related to reaction-diffusion equations. First, as ε→ 0the density ρε solution to (1.1) formally converges to a solution of the Fisher-KPP equation [11]:

∂tρ0(t, x)− ∂xxρ0(t, x) = F (ρ0(t, x)) .

The long time behaviour of such equation is well understood since the pioneering works byKolmogorov-Petrovsky-Piskunov [31] and Aronson-Weinberger [5]. For nonincreasing initial datawith sufficient decay at infinity the solution behaves asymptotically as a traveling front movingat the speed s = 2

√F ′(0). Moreover the traveling front solution with minimal speed is stable in

some L2 weighted space [23].In this work we prove that analogous results hold true in the parabolic regime ε2F ′(0) < 1.

Namely there exists a continuum of speeds [s∗(ε), ε−1) for which (1.1) admits smooth travelingfronts. The minimal speed is given by [16]

s∗(ε) =2√F ′(0)

1 + ε2F ′(0), if ε2F ′(0) < 1 . (1.5)

2

Obviously we have s∗(ε) ≤ min(2√F ′(0), ε−1). There also exists supersonic traveling fronts, with

speed s > ε−1. This appears surprising at first glance since the speed of propagation for thehyperbolic equation (1.1) is ε−1 (see formulation (1.3) and Section 2). These fronts are essentiallydriven by growth, since they travel faster than the maximum speed of propagation. The resultsare summarized in the following Theorem.

Theorem 1 (Parabolic regime). Assume that ε2F ′(0) < 1. The following alternatives hold:

(a) There exists no smooth or weak traveling front of speed s ∈ [0, s∗(ε)).

(b) For all s ∈ [s∗(ε), ε−1), there exists a smooth traveling front solution of (1.1) with speed s.

(c) For s = ε−1 there exists a weak traveling front.

(d) For all s ∈ (ε−1,∞) there also exists a smooth traveling front of speed s.

We also obtain that the minimal speed traveling front is nonlinearly locally stable in theparabolic regime ε2F ′(0) < 1 (see Section 5, Theorem 15).

There is a transition occuring when ε2F ′(0) = 1. In the hyperbolic regime ε2F ′(0) ≥ 1 theminimal speed speed becomes:

s∗(ε) = ε−1 , if ε2F ′(0) ≥ 1 . (1.6)

On the other hand, the front traveling with minimal speed s∗(ε) is discontinuous as soon asε2F ′(0) > 1. In the critical case ε2F ′(0) = 1 there exists a continuous but not smooth travelingfront with minimal speed s∗ =

√F ′(0).

Theorem 2 (Hyperbolic regime). Assume that ε2F ′(0) ≥ 1. The following alternatives hold:

(a) There exists no smooth or weak traveling front of speed s ∈ [0, s∗(ε)).

(b) There exists a weak traveling front solution of (1.1) with speed s∗(ε) = ε−1. The wave isdiscontinuous if ε2F ′(0) > 1.

(c) For all s ∈ (ε−1,∞) there exists a smooth traveling front of speed s.

We conclude this introduction by giving the precise definition of traveling fronts (smooth andweak) that will be used throughout the paper.

Definition 3. We say that a function ρ(t, x) is a smooth traveling front solution with speed sof equation (1.1) if it can be written ρ(t, x) = ν(x − st), where ν ∈ C2(R), ν ≥ 0, ν(−∞) = 1,ν(+∞) = 0 and ν satisfies

(ε2s2 − 1)ν′′(z)−(1− ε2F ′(ν(z))

)sν′(z) = F (ν(z)) , z ∈ R . (1.7)

We say that ρ is a weak traveling front with speed s if it can be written ρ(t, x) = ν(x− st), whereν ∈ L∞(R), ν ≥ 0, ν(−∞) = 1, ν(+∞) = 0 and ν satisfies (1.7) in the sense of distributions:

∀ϕ ∈ D(R) ,

∫R

((ε2s2 − 1)νϕ′′ +

(ν − ε2F (ν)

)sϕ′ − F (ν)ϕ

)dx = 0 .

In the following Section 2 we show some numerical simulations in order to illustrate our results.Section 3 is devoted to the proof of existence of the traveling fronts in the various regimes (resp.parabolic, hyperbolic, and supersonic). Finally, in Section 4 and Section 5 we prove the stabilityof the traveling fronts having minimal speed s∗(ε). We begin with linear stability (Section 4) sinceit is technically better tractable, and it let us discuss the case of the hyperbolic regime. We provethe full nonlinear stability in the range ε ∈ (0, 1/

√F ′(0)) (parabolic regime) in Section 5.

3

2 Numerical simulations

In this Section we perform numerical simulations of (1.1). We choose a logistic reaction term:F (ρ) = ρ(1 − ρ). We first symmetrize the hyperbolic system (1.3) by introducing f+ = 1

2 (ρ + j)and f− = 1

2 (ρ− j). This results in the following system:∂tf

+(t, x) + ε−1∂xf+(t, x) =

ε−2

2(f−(t, x)− f+(t, x)) +

1

2F (ρ(t, x))

∂tf−(t, x)− ε−1∂xf−(t, x) =

ε−2

2(f+(t, x)− f−(t, x)) +

1

2F (ρ(t, x)) .

(2.1)

In other words, the population is split into two subpopulations: ρ = f+ + f−, where the densityf+ denotes particles moving to the right with velocity ε−1, whereas f− denotes particles movingto the left with the opposite velocity.

We discretize the transport part using a finite volume scheme. Since we want to catch discon-tinuous fronts in the hyperbolic regime ε2F ′(0) > 1, we aim to avoid numerical diffusion. Thereforewe use a nonlinear flux-limiter scheme [26, 12]. The reaction part is discretized following the Eulerexplicit method.

f+n+1,i = f+n,i − ε−1 ∆t

∆x

(f+n,i + pi

∆x

2− f+n,i−1 − pi−1

∆x

2

)+ ε−2

∆t

2

(f−n,i − f

+n,i

)+

∆t

2F (ρn,i) .

The non-linear reconstruction of the slope is given by

pi = minmod

(f+n,i − f

+n,i−1

∆x,f+n,i+1 − f

+n,i

∆x

), minmod (p, q) =

{0 if sign (p) 6= sign (q)

min(|p|, |q|)sign (p) if sign (p) = sign (q)

We compute the solution on the interval (a, b) with the following boundary conditions: f+(a) = 1/2and f−(b) = 0. The discretization of the second equation for f− (2.1) is similar. The CFLcondition reads ∆t < ε∆x. It degenerates when ε ↘ 0, but we are mainly interested in thehyperbolic regime when ε is large enough. Other strategies should be used in the diffusive regimeε� 1, e.g. asymptotic-preserving schemes (see [19, 9] and references therein).

Results of the numerical simulations in various regimes (parabolic and hyperbolic) are shownin Figure 1.

3 Traveling wave solutions: Proof of Theorems 1 and 2

3.1 Characteristic equation

We begin with a careful study of the linearization of (1.7) around ν ≈ 0. We expect an exponentialdecay e−λz as z → +∞. The characteristic equation reads as follows,

(ε2s2 − 1)λ2 + (1− ε2F ′(0))sλ− F ′(0) = 0 . (3.1)

The discriminant is ∆ =(ε2F ′(0) + 1

)2s2 − 4F ′(0). Hence we expect an oscillatory behaviour in

the case ∆ < 0, i.e. s < s∗(ε). We assume henceforth s ≥ s∗(ε). In the case s < ε−1 (subsonicfronts) we have to distinguish between the parabolic regime ε2F ′(0) < 1 and the hyperbolic regimeε2F ′(0) > 1. In the former regime equation (3.1) possesses two positive roots, accounting for adamped behaviour. In the latter regime equation (3.1) possesses two negative roots. In the cases > ε−1 (supersonic fronts) we get two roots having opposite signs.

Next we investigate the linear behaviour close to ν ≈ 1. We expect an exponential relaxation1− eλ′z as z → −∞. The characteristic equation reads as follows,

(ε2s2 − 1)λ′2 − (1− ε2F ′(1))sλ′ − F ′(1) = 0 . (3.2)

4

0 10 20 30 400

0.2

0.4

0.6

0.8

(a) ε = 1/2

0 5 10 15 200

0.2

0.4

0.6

0.8

(b) ε = 1

0 5 10 15 200

0.2

0.4

0.6

0.8

(c) ε = 2

Figure 1: Numerical simulations of the equation (1.1) for F (ρ) = ρ(1− ρ) and for different valuesof ε = 0.5, 1, 2. Numerical method is described in Section 2. The initial data is the step functionf+(x < 0) = 1, f+(x > 0) = 0, and f− ≡ 0. For each value of ε we plot the density functionρ = f+ +f− in the (x, t) space, and the density ρ(t0, ·) at some chosen time t0. We clearly observein every cases a front traveling asymptotically at speed s∗(ε) as expected. We also observe thetransition between a smooth front and a discontinuous one. The transition occurs at ε = 1. In thecase ε = 1 we have superposed the expected profile ν(z) =

(1− ez/2

)+

in black, continous line,for the sake of comparison.

5

s < ε−1 s > ε−1

parabolic

if s < s∗(ε), NO

−0.2 0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

if s ≥ s∗(ε), YES

−0.2 0 0.2 0.4 0.6 0.8 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

YES

−0.2 0 0.2 0.4 0.6 0.8 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

hyperbolic

NO

−0.2 0 0.2 0.4 0.6 0.8 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

YES

−0.2 0 0.2 0.4 0.6 0.8 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Table 1: Phase plane dynamics depending on the regime (parabolic vs. hyperbolic) and the valueof the speed with respect to s∗(ε) and ε−1. In every picture the red line represents the travelingfront trajectory, and the green lines are the axes {u = 0} and {v = 0}. We do not consider thecase s = ε−1 since the dynamics are singular in this case and should be considered separately (seeSection 3.5).

We have ∆′ = [ε2F ′(1) + 1]2s2 − 4F ′(1) > 0. In the case s < ε−1 equation (3.2) possesses tworoots having opposite signs. In the case s > ε−1 it has two positive roots.

We summarize our expectations about the possible existence of nonnegative traveling fronts inTable 1.

3.2 Proof of Theorems 1.(a) and 2.(a): Obstruction for s < s∗(ε)

In this section we prove that no traveling front solution exists if the speed is below s∗(ε).

Proposition 4. There exists no traveling front with speed s for s < s∗(ε), where s∗(ε) is givenby (1.5)-(1.6).

Remark 5. Note that the proof below works in both cases ε2F ′(0) < 1 and ε2F ′(0) ≥ 1.

Proof. We argue by contradiction. The obstruction comes from the exponential decay at +∞.Assume that there exists such a traveling front ν(z). As s < s∗(ε), one has s < ε−1 in theparabolic as well as in the hyperbolic regime. Hence, as ν is bounded and satisfies the ellipticequation (1.7) in the sense of distributions, classical regularity estimates show that ν is smooth.It is necessarily decreasing as soon at it is below 1. Otherwise, it would reach a local minimumat some point z0 ∈ R, for which ν(z0) < 1, ν′(z0) = 0 and ν′′(z0) ≥ 0. It would then follow from(1.7) that F (ν(z0)) ≤ 0 and thus ν(z0) = 0. As F ∈ C1([0, 1]), the Cauchy-Lipschitz theoremwould imply ν ≡ 0, a contradiction.

Next, we define the exponential rate of decay at +∞:

λ := lim infz→+∞

−ν′(z)ν(z)

≥ 0 .

Consider a sequence zn → +∞ such that −ν′(zn)/ν(zn)→ λ and define the renormalized shift:

νn(z) :=ν(z + zn)

ν(zn).

6

This function is locally bounded by classical Harnack estimates. It satisfies

(ε2s2 − 1)ν′′n(z) +(ε2F ′(ν(z + zn))− 1

)sν′n(z) =

1

ν(zn)F (ν(zn)νn(z)) , z ∈ R .

As F ∈ C1([0, 1]), F (0) = 0 and F is concave, the functions z 7→(ε2F ′(ν(z + zn))− 1

)s and z 7→

1ν(zn)

F (ν(zn)νn(z)) are uniformly bounded, uniformly in n. Hence, Schauder elliptic regularity

estimates yield that the sequence (νn)n is locally bounded in the Holder space Cα(K) for anycompact subset K ⊂ R and any α ∈ (0, 1). The Ascoli theorem and a diagonal extraction processgive an extraction, that we still denote (νn)n, such that (νn)n converges to some function ν∞ inCα(K) for any compact subset K ⊂ R and any α ∈ (0, 1). The limiting function is a solution inthe sense of distributions of

(ε2s2 − 1)ν′′∞(z) +(ε2F ′(0)− 1

)sν′∞(z) = F ′(0)ν∞(z) , z ∈ R . (3.3)

As this equation is linear, one has ν∞ ∈ C∞(R). If ν∞(z0) = 0, then as ν∞ is nonnegative, onewould get ν′∞(z0) = 0 and thus ν∞ ≡ 0 by uniqueness of the Cauchy problem, which would be acontradiction since ν∞(0) = limn→+∞ νn(0) = 1. Thus ν∞ is positive.

Define V = ν′∞/ν∞. The definition of λ yields minR V = V (0) = −λ. Thus V ′(0) = 0. Hencewe deduce from (3.3) that ν∞(z) = ν∞(0)e−λz. Plugging this into (3.3), we obtain that λ satisfiesthe following second order equation,

(ε2s2 − 1)λ2 − (ε2F ′(0)− 1)sλ− F ′(0) = 0 .

We know from Section 3.1 that, both in the parabolic and hyperbolic regimes, there is no real rootin the case s < s∗(ε).

3.3 Proof of Theorem 1.(b): Existence of smooth traveling fronts in theparabolic regime s ∈ [s∗(ε), ε−1)

In [27] the author proves the existence of traveling front, by reducing the problem to the classicalFisher-KPP problem. It is required that the nonlinear function 1−ε2F ′(ρ) remains positive, whichreads exactly ε2F ′(0) < 1 in our context. We present below a direct proof based on the methodof sub- and supersolutions, following the method developed by Berestycki and Hamel in [6].

3.3.1 The linearized problem

Proposition 6. Let λs be the smallest (positive) root of the characteristic polynomial (3.1). Thenν(z) = min{1, e−λsz} is a supersolution of (1.7).

Proof. Let r(z) = e−λsz. Then as r is decreasing and F is concave, it is easy to see that r is asupersolution of (1.7). On the other hand, the constant function 1 is clearly a solution of (1.7).We conclude since the minimum of two supersolutions is a supersolution.

3.3.2 Resolution of the problem on a bounded interval

Proposition 7. For all a > 0 and τ ∈ R, there exists a solution νa,τ of(ε2s2 − 1)ν′′a,τ + (ε2F ′(νa,τ )− 1)sν′a,τ = F (νa,τ ) in (−a, a),νa,τ (−a) = ν(−a+ τ),νa,τ (a) = ν(a+ τ).

(3.4)

Moreover, this function is nonincreasing over (−a, a) and it is unique in the class of nonincreasingfunctions.

In order to prove this result, we consider the following sequence of problems:

7

• ν0(z) = ν(z + τ)

• νn+1 is solution to(ε2s2 − 1)ν′′n+1 + (ε2F ′(νn)− 1)sν′n+1 +Mνn+1 = F (νn) +Mνn in (−a, a),

νn+1(−a) = ν(−a+ τ),

νn+1(a) = ν(a+ τ),

(3.5)

where ν is defined in Proposition 6 and M > s2

2

(ε2F ′(0) − 1

)is large enough so that

s 7→ F (s) +Ms is increasing.

Lemma 8. The sequence (νn)n is well-defined. The functions z 7→ νn(z) are nonincreasing andfor all z ∈ (−a, a), the sequence (νn(z))n is nonincreasing.

Proof. We prove this Lemma by induction. Clearly, ν0 is nonincreasing. First, one can find aunique weak solution ν1 ∈ C0([−a, a]) of

(ε2s2 − 1)ν′′1 + (ε2F ′(ν0)− 1)sν′1 +Mν1 = F (ν0) +Mν0 in (−a, a),

ν1(−a) = ν(−a+ τ),

ν1(a) = ν(a+ τ),

(3.6)

using the Lax-Milgram theorem and noticing that the underlying operator is coercive since M >s2

2

(ε2F ′(0)− 1

)and s < ε−1.

Let w0 = ν1 − ν0. As ν0 is a supersolution of equation (1.7), one has{(ε2s2 − 1)w′′0 + (ε2F ′(ν0)− 1)sw′0 +Mw0 ≤ 0 in (−a, a),w0(−a) = w0(a) = 0.

As M > 0, the weak maximum principle gives w0 ≤ 0, that is, ν1 ≤ ν0.Define the constant function ν = ν(a+ τ). It satisfies

(ε2s2 − 1)ν′′ + (ε2F ′(ν0)− 1)sν′ +Mν = Mν ≤ F (ν) +Mν ≤ F (ν0) +Mν0

in (−a, a) since s 7→ F (s) +Ms is increasing and ν0(z) = ν(z+ τ) ≥ ν(a+ τ) = ν by monotonicityof ν. The same arguments as above lead to ν1 ≥ ν.

Assume that Lemma 8 is true up to rank n. The existence and the uniqueness of νn+1 followfrom the same arguments as that of ν1. Let wn = νn+1 − νn. As F is concave and νn−1 ≥ νn, weknow that F ′(νn−1) ≤ F ′(νn). As νn is nonincreasing, we thus get{

(ε2s2 − 1)w′′n + (ε2F ′(νn)− 1)sw′n +Mwn ≤ 0 in (−a, a),wn(−a) = wn(a) = 0.

Hence, wn ≤ 0 and thus νn+1 ≤ νn. Similarly, one easily proves that νn+1 ≥ ν in (−a, a).Differentiating (3.5) and denoting v = ν′n+1, one gets

(ε2s2 − 1)v′′ +(ε2F ′(ν0)− 1

)sv′ +

(M + ε2F ′′(ν0)ν′0

)v =

(F ′(ν0) +M

)ν′0 ≤ 0 in (−a, a)

since s 7→ F (s)+Ms is increasing and ν0 is nonincreasing. As F is concave, the zeroth-order termis positive and thus the elliptic maximum principle ensures that v reaches its maximum at z = −aor at z = a. But as ν(a+ τ) ≤ νn+1(z) ≤ ν(z + τ) for all z ∈ (−a, a), one has

v(−a) ≤ lim supz→−a+

νn+1(z)− νn+1(−a)

z + a≤ lim sup

z→−a+

ν(z + τ)− ν(−a+ τ)

z + a≤ 0

and similarly v(a) ≤ 0. Thus v ≤ 0, meaning that νn+1 is nonincreasing.

8

Proof of Proposition 7. As the sequence (νn)n is decreasing and bounded from below, it admits alimit νa,τ as n → +∞. It easily follows from the classical regularity estimates that νa,τ satisfiesthe properties of Proposition 7.

If ν1 and ν2 are two nondecreasing solutions of (3.4), then the same arguments as before givethat νµ1 < ν1 in Σµ for all µ ∈ (0, 2a). Hence, ν1 ≤ ν2 and a symmetry argument gives ν1 ≡ ν2.

Lemma 9. For all a > 0, there exists τa ∈ R such that νa,τa(0) = 12 .

Proof. Define I(τ) := νa,τ (0). It follows from the classical regularity estimates and from theuniqueness of νa,τ that I is a continuous function. Moreover, as νa,τ is nonincreasing, one has

ν(a+ τ) ≤ I(τ) ≤ ν(−a+ τ),

where ν is defined in Proposition 6. As ν(· + τ) → 0 as τ → +∞ and ν(· + τ) → 1 as τ → −∞locally uniformly on R, one has I(−∞) = 1 and I(+∞) = 0. The conclusion follows.

3.3.3 Existence of traveling fronts with speeds s ∈ [s∗(ε), ε−1)

We conclude by giving the proof of Theorem 1 as a combination of the above results.

Proof of Theorem 1. Consider a sequence (an)n such that limn→+∞ an = +∞ and define νn(z) :=νan,τan

for all z ∈ [−an, an]. This function is decreasing and satisfies νn(0) = 1/2, 0 ≤ νn ≤ 1 and

(ε2s2 − 1)ν′′n + (ε2F ′(νn)− 1)sν′n = F (νn) in (−an, an).

As in the proof of Proposition 4, the uniform boundedness of (νn)n together with Lp ellipticregularity estimates ensure that the sequence (νn)n is uniformly bounded in W 2,p(K) for allcompact set K ∈ R and p ∈ (1,∞). It follows from Sobolev injections and the Ascoli theoremthat the sequence (νn)n converges in C0loc(R) as n→ +∞ to a function ν, up to extraction. Thenν satisfies

(ε2s2 − 1)ν′′ + (ε2F ′(ν)− 1)sν′ = F (ν) .

Moreover it is nonincreasing, 0 ≤ ν ≤ 1 and ν(0) = 1/2.Define `± := limz→±∞ ν(z). Passing to the (weak) limit in the equation satisfied by ν, one

gets F (`±) = 0. As 0 ≤ `± ≤ 1, the hypotheses on F give `± ∈ {0, 1}. On the other hand, as ν isnonincreasing, one has

`+ ≤ ν(0) = 1/2 ≤ `−.

We conclude that `− = ν(−∞) = 1 and `+ = ν(+∞) = 0.

The following classical inequality satisfied by the traveling profile will be required later.

Lemma 10. The traveling profile ν satisfies: ∀z ν′(z)+λν(z) ≥ 0, where λ is the smallest positiveroot of (3.1).

Proof. We introduce ϕ(z) = −ν′(z)ν(z) . It is nonnegative, and it satisfies the following first-order

ODE with a source term(ε2s2 − 1

) (−ϕ′(z) + ϕ(z)2

)+(1− ε2F ′(ν(z))

)sϕ(z) =

F (ν(z))

ν(z).

Since F is concave, ϕ satisfies the differential inequality(1− ε2s2

)ϕ′(z) ≤

(1− ε2s2

)ϕ(z)2 −

(1− ε2F ′(0)

)sϕ(z) + F ′(0) .

The right-hand-side is the characteristic polynomial of the linearized equation (3.1). Moreover thefunction ϕ verifies limz→−∞ ϕ(z) = 0. Hence a simple ODE argument shows that ∀z ϕ(z) ≤ λ.

9

3.4 Proof of Theorem 1.(c): Existence of weak traveling fronts of speeds = ε−1 in the parabolic regime

The aim of this Section is to prove that in the parabolic regime ε2F ′(0) < 1, there still existstraveling fronts in the limit case s = ε−1 but in the weak sense.

Proposition 11. Assume that ε2F ′(0) < 1. Then there exists a weak traveling front of speeds = ε−1.

Proof. Let sn = ε−1 − 1/n for all n large enough so that sn ≥ s∗(ε). We know from the previousSection that we can associate with the speed sn a smooth traveling front νn and that we canassume, up to translation, that νn(0) = 1/2. Multiplying equation (1.7) by ν′n and integrating byparts over R, one gets

sn(1− ε2F ′(0)

) ∫Rν′n(z)2dz ≤ sn

∫R

(1− ε2F ′

(νn(z)

))ν′n(z)2dz

= −∫RF(νn(z)

)ν′n(z)dz

= −∫ 1

0

F (u)du.

Hence, as ε2F ′(0) < 1, the sequence (ν′n)n is bounded in L2(R) and one can assume, up toextraction, that it admits a weak limit V∞ in L2(R). It follows that the sequence (νn)n convergeslocally uniformly to ν∞(z) :=

∫ z0V∞(z′)dz′ + 1/2. Passing to the limit in (1.7), we get that this

function is a weak solution of

−(1− ε2F ′(ν∞(z))

)sν′∞(z) = F (ν∞(z)) , z ∈ R ,

which ends the proof.

3.5 Proof of Theorem 2.(b): Existence of weak traveling fronts of speeds = ε−1 in the hyperbolic regime

In this Section we investigate the existence of traveling fronts with critical speed s = ε−1 in thehyperbolic regime ε2F ′(0) = 1.

Proof of Theorem 2. The function G(ρ) := ε2F (ρ) − ρ is concave, and vanishes when ρ = 0.Furthermore, G(1) < 0 and G′(0) = ε2F ′(0) − 1 ≥ 0. We now distinguish between the two casesε2F ′(0) > 1 and ε2F ′(0) = 1.

1. First case: ε2F ′(0) > 1. As G′ is decreasing, there exists a unique θε ∈ (0, 1) such that Gvanishes.

2. Second case: ε2F ′(0) = 1. The only root of G is ρ = 0. In this case we set θε = 0.

For both cases, we have G′(ρ) < 0 for all ρ > θε since G is strictly concave and G(0) = G(θε) = 0.Hence, ε2F ′(ρ) < 1 for all ρ > θε. Set ν the maximal solution of

ν′(z) =εF (ν(z))

ε2F ′(ν(z))− 1,

ν(0) =1 + θε

2> θε.

(3.7)

Let I be the (maximal) interval of definition of ν, with 0 ∈ I, and

z0 = sup{z ∈ I, ν(z) > θε}.

10

1- Conclusion of the argument in the first case: ε2F ′(0) > 1.

Since θε > 0, we have necessarily z0 < +∞. From (3.7), ν is decreasing on (−∞, z0). Thus, wehave ν(z)→ θε as z → z−0 . Moreover, one easily gets ν(−∞) = 1.

We set ν(z0) = θε and we extend ν by 0 over (z0,∞). We observe that ν is a weak solution, inthe sense of distributions, of (

ε2F (ν)− ν)′

= εF (ν) on R

since ε2F (0) = 0 and ε2F (θε) = θε.Up to space shifting z − z0, we may assume that the discontinuity arises at z = 0.

Example: the case F (ρ) = ρ(1− ρ) and ε > 1. The traveling profile solves

ν′(z) =εν(z)(1− ν(z))

ε2 − 1− 2ε2ν(z),

or equivalently

ν(z)ε2−1 (1− ν(z))

ε2+1= keεz .

The constant k is determined by the condition ν(0) = θε = 1 − ε−2. Finally the traveling profileν(z) satisfies the following implicit relation:

ν(z)ε2−1 (1− ν(z))

1+ε2=(1− ε−2

)ε2−1 (ε−2)ε2+1

eεz =(ε2 − 1

)ε2−1eεz+2ε2 log ε2 . (3.8)

2- Conclusion of the argument in the second case: ε2F ′(0) = 1.

The difference here is that θε = 0. To conclude the proof as previously, we just need to check thatz0 is finite. We argue by contradiction. Assume z0 = +∞. Linearizing the r.h.s. of (3.7) nearν = 0, we get

ν′(z) =F ′(0)

εF ′′(0)+ o(ν(z)) , as z → +∞ (3.9)

We get a contradiction because ε−1F ′(0)/F ′′(0) < 0.Finally, we create a continuous front with the same extension idea as for the first case.

Example: the case F (ρ) = ρ(1− ρ) and ε = 1. The traveling profile reads (3.8):

ν(z) =(

1− ez/2)+.

3.6 Proof of Theorem 1.(d) and Theorem 2.(c): Existence of supersonictraveling fronts s > ε−1

In this Section we investigate the existence of supersonic traveling fronts with speeds above themaximal speed of propagation s > ε−1. These fronts are essentially driven by growth. Theexistence of such ”unrealistic” fronts is motivated by the extreme case ε→ +∞ for which we haveformally ∂tρ = F (ρ) (1.3). There exist traveling fronts of arbitrary speed which are solutions to−sν′ = F (ν).

Proposition 12. Given any speed s > ε−1 there exists a smooth traveling front ν(x − st) withthis speed.

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

Figure 2: Supersonic traveling front in the phase plane (V, V ′) for the nonlinearity F (ρ) = ρ(1−ρ),and parameters ε =

√2, s = 1 > ε−1. Be aware of the time reversal ν(z) = V (−z), which is the

reason why V ′ ≥ 0. The red line represents the traveling profile, and the green line represents thesupersolution kF (v).

Proof. We sketch the proof. We give below the key arguments derived from phase plane analysis.The same procedure as developped in Section 3.3 based on sub- and supersolutions could bereproduced based on the following ingredients.

We learn from simple phase plane considerations associated to (1.7) that the situation is re-versed in comparison to the classical Fisher-KPP case (or ε2F ′(0) < 1 and s ∈ [s∗(ε), ε−1)).Namely the point (0, 0) is a saddle point (instead of a stable node) whereas (1, 0) is an unstablenode (instead of saddle point). This motivates ”time reversal”: V (z) = ν(−z). Equation (1.7)becomes

(ε2s2 − 1)V ′′(z)−(ε2F ′ (V (z))− 1

)sV ′(z) = F (V (z)) , z ∈ R .

We make the classical phase-plane transformation V ′ = P [31, 20]. We end up with the implicitODE with Dirichlet boundary conditions for P :

(ε2s2 − 1)P ′(v)−(ε2F ′ (v)− 1

)s =

F (v)

P (v), P (0) = P (1) = 0 .

The unstable direction is given by P (v) = λv where λ is the positive root of

(ε2s2 − 1)λ−(ε2F ′ (0)− 1

)s =

F ′(0)

λ. (3.10)

Since F is concave we deduce that P (v) = λv is a supersolution as in Proposition 6. In fact,denoting Q(v) = P (v)− λv we have

(ε2s2 − 1)Q′(v) = (ε2s2 − 1)(P ′(v)− λ) ≤ sε2 (F ′(v)− F ′(0)) +F (v)

P (v)− F ′(0)

λ,

≤ F ′(0)v

(1

P (v)− 1

λv

)≤ − F

′(0)

λP (v)Q(v) .

Hence the trajectory leaving the saddle point (0, 0) in the phase plane (V, V ′) remains below theline V ′ ≤ λV .

12

On the other hand it is straightforward to check that kF (v) is a supersolution where k =ε2s/(ε2s2 − 1). We denote R(v) = P (v)− kF (v). We have ks > 1 and

(ε2s2 − 1)R′(v) = (ε2s2 − 1)(P ′(v)− kF ′(v)) = ε2sF ′(v)− s+F (v)

P (v)− (ε2s2 − 1)kF ′(v)

= −s+1

k− R(v)

kP (v)< − R(v)

kP (v).

We also show that initially (as v → 0) we have kF ′(0) > λ. This proves that R(v) ≤ 0 for allv ∈ (0, 1). Indeed, we plug kF ′(0) in place of λ into (3.10) and we get

(ε2s2 − 1)kF ′(0)−(ε2F ′ (0)− 1

)s = s >

1

k=

F ′(0)

kF ′(0).

As a conclusion the trajectory leaving the saddle node at (0, 0) is trapped in the set {0 ≤v ≤ 1 , 0 ≤ p ≤ kF (v)} (see Fig. 2). By the Poincare-Bendixon Theorem it necessarily convergesto the stable node at (1, 0). This heteroclinic trajectory is the traveling front in the supersoniccase.

4 Linear stability of traveling front solutions

In this Section we investigate the linear stability of the traveling front having minimal speeds = s∗(ε) in both the parabolic and the hyperbolic regime. We seek stability in some weightedL2 space. The important matter here is to identify the weight eφ. The same weight shall be usedcrucially for the nonlinear stability analysis (Section 5).

We recall that the minimal speed is given by

s∗(ε) =

2√F ′(0)

1 + ε2F ′(0)if ε2F ′(0) < 1

ε−1 if ε2F ′(0) ≥ 1

The profile of the wave has the following properties in the case ε2F ′(0) < 1:

∀z ν(z) ≥ 0 , ∂zν(z) ≤ 0 , ∂zν(z) + λν(z) ≥ 0 ,

where the decay exponent λ is

λ =s(1− ε2F ′(0))

2(1− ε2s2)=

1 + ε2F ′(0)

1− ε2F ′(0).

We will use in this Section the formulation (1.3) of our system. The linearized system aroundthe stationary profile ν in the moving frame z = x− st reads (∂t − s∂z)u+ ∂z

(vε

)= F ′(ν)u

ε(∂t − s∂z)v + ∂zu = −vε.

(4.1)

Proposition 13. Let ε > 0. In the hyperbolic regime ε2F ′(0) ≥ 1 assume in addition that theinitial perturbation has the same support as the wave. There exists a function φε(z) such that theminimal speed traveling front is linearly stable in the weighted L2(e2φε(z)dz) space. More preciselythe following Lyapunov identity holds true for solutions of the linear system (4.1),

d

dt

(1

2

∫R

(|u|2 + |v|2

)e2φε(z) dz

)≤ 0 .

13

Proof. We denote φ = φε for the sake of clarity. We multiply the first equation by ue2φ, and thesecond equation by ve2φ, where φ is to be determined. We get

d

dt

(1

2

∫R|u|2e2φ(z) dz

)+s

2

∫R|u|2∂ze2φ(z) dz +

∫R∂z

(vε

)ue2φ(z) dz =

∫RF ′(ν)|u|2e2φ(z) dz ,

d

dt

(1

2

∫R|v|2e2φ(z) dz

)+s

2

∫R|v|2∂ze2φ(z) dz +

∫R∂z

(uε

)ve2φ(z) dz = − 1

ε2

∫R|v|2e2φ(z) dz .

Summing the two estimates we obtain

d

dt

(1

2

∫R

(|u|2 + |v|2

)e2φ(z) dz

)+

∫R

(s∂zφ(z)− F ′(ν)) |u|2e2φ(z) dz+∫R

(s∂zφ(z) +

1

ε2

)|v|2e2φ(z) dz−2

ε

∫R

(∂zφ(z))uve2φ(z) dz = 0 .

We seek an energy dissipation estimate, see (4.3) below. Therefore we require that the lastquadratic form acting on (u, v) is nonnegative. This is guaranteed if ∂zφ ≥ 0 and the followingdiscriminant is nonpositive:

∆(z) =4

ε2(∂zφ(z))

2 − 4 (s∂zφ(z)− F ′(ν))

(s∂zφ(z) +

1

ε2

)=

4

ε2

((1− ε2s2

)(∂zφ(z))

2 − s(1− ε2F ′(ν)

)∂zφ(z) + F ′(ν)

). (4.2)

The rest of the proof is devoted to finding such a weight φ(z) satisfying this sign condition. Wedistinguish between the parabolic and the hyperbolic regime.

1- The parabolic regime.

In the case ε2F ′(0) < 1 we have ε2s2 < 1. Hence the optimal choice for ∂zφ is:

∂zφ(z) =s(1− ε2F ′(ν)

)2 (1− ε2s2)

= λ1− ε2F ′(ν)

1− ε2F ′(0)≥ 0 .

Notice that ∂zφ→ λ as z → +∞. We check that the discriminant is indeed nonpositive:

ε2∆(z) = −4(1− ε2s2

)(∂zφ(z))

2+ 4F ′(ν)

=1

(1− ε2s2)

(−s2

(1 + ε2F ′(ν)

)2+ 4F ′(ν)

)=

1

(1− ε2F ′(0))2

(−4F ′(0)

(1 + ε2F ′(ν)

)2+ 4F ′(ν)

(1 + ε2F ′(0)

)2)=

−4

(1− ε2F ′(0))2 (F ′(0)− F ′(ν))

(1− ε4F ′(0)F ′(ν)

).

We have ∆(z) ≤ 0 since ∀z F ′(ν(z)) ≤ F ′(0) and ε2F ′(0) < 1. Since the quadratic form isnonnegative, we may control it by a sum of squares. This is the purpose of the next computation.We have∣∣∣∣2ε

∫R

(∂zφ(z))uve2φ(z) dz

∣∣∣∣≤∫R

(s∂zφ(z)− F ′(ν)−A(z)) |u|2e2φ(z) dz +

∫R

(s∂zφ(z) +

1

ε2−A(z)

)|v|2e2φ(z) dz ,

where A(z) is solution of

4 (s∂zφ(z)− F ′(ν)−A(z))

(s∂zφ(z) +

1

ε2−A(z)

)=

4

ε2(∂zφ(z))

2.

14

A straightforward computation gives

2A(z) =

(2s∂zφ(z)− F ′(ν) +

1

ε2

)−

((F ′(ν) +

1

ε2

)2

+4

ε2(∂zφ(z))

2

)1/2

=1− ε2F ′(ν)

ε2 (1− ε2s2)− 1

ε2 (1− ε2s2)

((1− ε2s2

)2 (1 + ε2F ′(ν)

)2+ ε2s2

(1− ε2F ′(ν)

)2)1/2=

1− ε2F ′(ν)

ε2 (1− ε2s2)

1−

((1− ε2F ′(0)

1 + ε2F ′(0)

)4(1 + ε2F ′(ν)

1− ε2F ′(ν)

)2

+4ε2F ′(0)

(1 + ε2F ′(0))2

)1/2

=1− ε2F ′(ν)

ε2 (1− ε2s2)

1−

(1 +

(1− ε2F ′(0)

1 + ε2F ′(0)

)4(1 + ε2F ′(ν)

1− ε2F ′(ν)

)2

−(1− ε2F ′(0)

)2(1 + ε2F ′(0))

2

)1/2 .

We clearly have A(z) ≥ 0 since

∀z 1 + ε2F ′(ν)

1− ε2F ′(ν)≤ 1 + ε2F ′(0)

1− ε2F ′(0).

Finally we obtain in the case ε2F ′(0) < 1,

d

dt

(1

2

∫R

(|u|2 + |v|2

)e2φ(z) dz

)+

∫RA(z)

(|u|2 + |v|2

)e2φ(z) dz ≤ 0 . (4.3)

2- The hyperbolic regime.

We assume for simplicity that the support of the traveling profile is Supp ν = (−∞, 0].In the hyperbolic regime we have s = ε−1, so the discriminant equation (4.2) reduces to

∆(z) =4

ε2(−s(1− ε2F ′(ν)

)∂zφ(z) + F ′(ν)

).

We naturally choose

∂zφ(z) =εF ′(ν)

1− ε2F ′(ν).

Within this choice for φ we get,

d

dt

(1

2

∫z≤0

(|u|2 + |v|2

)e2φ(z) dz

)+

∫z≤0

A(z)(ε2F ′(ν(z))u− v

)2e2φ(z) dz = 0 ,

where the additional weight in the dissipation writes:

A(z) =1

ε2 (1− ε2F ′(ν(z))).

In the case ε2F ′(0) > 1 we have 1 − ε2F ′(ν(z)) > 0 on Supp ν (see Section 3.5). Notice thatthe monotonicity of φ may change on Supp ν since F ′(ν(z)) may change sign. We observe thatA(z) is uniformly bounded from below on Supp ν.

In the transition case ε2F ′(0) = 1, we have ∂zφ(z) → +∞ as z → 0−. We observe thatA(z)→ +∞ as z → 0− too.

Example: the case F (ρ) = ρ(1− ρ), and ε = 1. We can easily compute from Section 3.5

φ(z) = −z2− log

(1− ez/2

).

15

Remark 14 (Lack of coercivity). 1- The parabolic regime. We directly observe that A(z)→ 0as z → +∞ in the Lyapunov identity (4.3). This corresponds to the lack of coercivity of the linearoperator. It has been clearly identified for the classical Fisher-KPP equation [23, 24]. This lack ofcoercivity is a source of complication for the next question, i.e. nonlinear stability (see Section 5).2- The hyperbolic regime. The situation is more degenerated here: the dissipation providesinformation about the relaxation of v towards ε2F ′(ν)u only.

5 Nonlinear stability of traveling front solutions in the parabolicregime ε2F ′(0) < 1

In this Section we investigate the stability of the traveling profile having minimal speed s = s∗(ε) inthe parabolic regime. We seek stability in the energy class. Energy methods have been successfullyapplied to reaction-diffusion equations [23, 24, 43, 25]. We follow the strategy developped in [24]for a simpler equation, namely the damped hyperbolic Fisher-KPP equation.

Before stating the theorem we give some useful notations. The perturbation is denoted byu(t, z) = ρ(t, z)− ν(z) where z = x− st is the space variable in the moving frame. We also needsome weighted perturbation w = eφu, where φ is an explicit weight to be precised later (5.11).

Theorem 15. For all ε ∈(

0, 1/√F ′(0)

)there exists a constant c(ε) such that the following claim

holds true: let u0 be any compactly supported initial perturbation which satisfies

‖u0‖2H1(R) + ‖w0‖2H1(R) ≤ c(ε) ,

then there exists z0 ∈ R such that

supt>0

(‖∂zu(t, ·)‖22 +

∫z<z0

|u(t, z)|2 dz + ‖w(t, ·)‖2H1

)≤ c(ε) ,

remains uniformly small for all time t > 0, and the perturbation is globally decaying in the followingsense: (

‖∂zu‖22 +

∫z<z0

|u|2 dz + ‖∂zw‖22 +

∫z>z0

e−φ(z)|w|2 dz)∈ L2(0,+∞) .

Remark 16. 1. The additional weight e−φ(z) in the last contribution (weighted L2 space) isspecific to the lack of coercivity in the energy estimates.

2. The constant c(ε) that we obtain degenerates as ε → 1/√F ′(0), due to the transition from

a parabolic to an hyperbolic regime.

3. We restrict ourselves to compactly supported initial perturbations u0 to justify all integrationby parts. Indeed the solution u(t, z) remains compactly supported for all t > 0 because ofthe finite speed of propagation (see the kinetic formulation (2.1) and [15, Chapter 12]). Theresult would be the same if we were assuming that u0 decays sufficiently fast at infinity.

Proof. We proceed in several steps.

1- Derivation of the energy estimates. The equation satisfied by the perturbation u writes

ε2(∂ttu− 2s∂tzu+ s2∂zzu

)+(1− ε2F ′(ν + u)

)(∂tu− s∂zu)− ∂zzu

+ ε2(F ′(ν + u)− F ′(ν))s∂zν = F (ν + u)− F (ν) . (5.1)

We write the nonlinearities as follows:

F ′(ν + u) = F ′(ν) +K1(z;u)u ,

F ′(ν + u)− F ′(ν) = F ′′(ν)u+K2(z;u)u2 ,

F (ν + u)− F (ν) = F ′(ν)u+K3(z;u)u2 .

16

where the functions Ki are uniformly bounded in L∞(R). More precisely we have

K1(z;u) =

∫ 1

0

F ′′(ν+tu)dt , K2(z;u) =

∫ 1

0

(1−t)F ′′′(ν+tu)dt , K3(z;u) =

∫ 1

0

(1−t)F ′′(ν+tu)dt .

Thus we can decompose equation (5.1) into linear and nonlinear contributions:

ε2(∂ttu− 2s∂tzu+ s2∂zzu

)+(1− ε2F ′(ν)

)(∂tu− s∂zu)− ∂zzu+

(sε2F ′′(ν)∂zν − F ′(ν)

)u

= ε2K1(z;u)u (∂tu− s∂zu) +(K3(z;u)− sε2K2(z;u)∂zν

)u2 . (5.2)

Testing equation (5.2) against ∂tu− s∂zu yields our first energy estimate (hyperbolic energy):

d

dt

{ε2

2

∫R|∂tu− s∂zu|2 dz +

1

2

∫R|∂zu|2 dz +

1

2

∫R

(sε2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz

}+

∫R

(1− ε2F ′(ν)

)|∂tu− s∂zu|2 +

s

2

∫R∂z(sε2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz

= ε2∫RK1(z;u)u |∂tu− s∂zu|2 dz +

∫R

(K3(z;u)− se2K2(z;u)∂zν

)u2 (∂tu− s∂zu) dz . (5.3)

We are lacking coercivity with respect to H1 norm in the energy dissipation. Testing equation(5.2) against u yields our second energy estimate (parabolic energy):

d

dt

{ε2∫Ru (∂tu− s∂zu) dz +

1

2

∫R

(1− ε2F ′(ν)

)|u|2 dz

}− ε2

∫R|∂tu− s∂zu|2 dz +

∫R|∂zu|2 dz +

∫R

(sε2

2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz

= ε2∫RK1(z;u)u2 (∂tu− s∂zu) dz +

∫R

(K3(z;u)− se2K2(z;u)∂zν

)u3 dz . (5.4)

We introduce the following notations for the two energy contributions and the respective quadraticdissipations (5.3),(5.4):

Eu1 (t) =ε2

2

∫R|∂tu− s∂zu|2 dz +

1

2

∫R|∂zu|2 dz +

1

2

∫R

(sε2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz ,

Eu2 (t) = ε2∫Ru (∂tu− s∂zu) dz +

1

2

∫R

(1− ε2F ′(ν)

)|u|2 dz ,

Qu1 (t) =

∫R

(1− ε2F ′(ν)

)|∂tu− s∂zu|2 +

s

2

∫R∂z(sε2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz ,

Qu2 (t) = −ε2∫R|∂tu− s∂z|2 dz +

∫R|∂zu|2 dz +

∫R

(sε2

2F ′′(ν)∂zν − F ′(ν)

)|u|2 dz .

The delicate issue is to control the zeroth-order terms. In particular we define the weights

A1(z) = sε2F ′′(ν)∂zν − F ′(ν) ,

A2(z) =sε2

2F ′′(ν)∂zν − F ′(ν) .

They change sign over R. More precisely we have

limz→−∞

A1(z) = limz→−∞

A2(z) = −F ′(1) > 0 ,

limz→+∞

A1(z) = limz→+∞

A2(z) = −F ′(0) < 0 .

17

To circumvent this issue we introduce w(t, z) = eφ(z)u(t, z) as in [24] and the previous Section4, where φ(z) is a weight to be determined later (5.10). The new function w(t, z) satisfies thefollowing equation:

ε2∂ttw − 2ε2s∂tzw +(2ε2s∂zφ+ 1− ε2F ′(ν)

)∂tw +

(−s(1− ε2F ′(ν)

)− 2(ε2s2 − 1)∂zφ

)∂zw

+ (ε2s2− 1)∂zzw+(s(1− ε2F ′(ν)

)∂zφ+ (ε2s2 − 1)

(−∂zzφ+ |∂zφ|2

)+ ε2sF ′′(ν)∂zν − F ′(ν)

)w

= ε2K1(z;u)u (∂tw − s∂zw) +(K3(z;u)− ε2sK2(z;u)∂zν + ε2sK1(z;u)∂zφ

)uw . (5.5)

We denote the prefactors of ∂tw, ∂zw and w as A3, A4 and A5 respectively:

A3(z) = 2ε2s∂zφ+ 1− ε2F ′(ν) ,

A4(z) = −s(1− ε2F ′(ν)

)− 2(ε2s2 − 1)∂zφ , (5.6)

A5(z) = s(1− ε2F ′(ν)

)∂zφ+ (ε2s2 − 1)

(−∂zzφ+ |∂zφ|2

)+ ε2sF ′′(ν)∂zν − F ′(ν) .

Testing (5.5) against ∂tw, we obtain our third energy estimate:

d

dt

{ε2

2

∫R|∂tw|2 dz +

1− ε2s2

2

∫R|∂zw|2 dz +

1

2

∫RA5(z)|w|2 dz

}+

∫RA3(z)|∂tw|2 dz +

∫RA4(z)∂tw∂zw dz

= ε2∫RK1(z;u)u

(|∂tw|2 − s∂tw∂zw

)dz+

∫R

(K3(z;u)− ε2sK2(z;u)∂zν + ε2sK1(z;u)∂zφ

)uw∂tw dz ,

Testing (5.5) against w we obtain our last energy estimate:

d

dt

{ε2∫Rw∂tw dz +

1

2

∫RA3(z)|w|2 dz

}− ε2

∫R|∂tw|2 dz + (1− ε2s2)

∫R|∂zw|2 dz + 2sε2

∫R∂tw∂zw dz +

∫R

(A5(z)− ∂zA4(z)

2

)|w|2 dz

=

∫R

(K3(z;u)− ε2sK2(z;u)∂zν + ε2sK1(z;u)∂zφ

)uw2 dz+

∫Rε2K1(z;u)uw (∂tw − s∂zw) dz ,

We introduce again useful notations for the two energy contributions and the associated quadraticdissipations:

Ew1 (t) =ε2

2

∫R|∂tw|2 dz +

1− ε2s2

2

∫R|∂zw|2 dz +

1

2

∫RA5(z)|w|2 dz , (5.7)

Ew2 (t) = ε2∫Rw∂tw dz +

1

2

∫RA3(z)|w|2 dz , (5.8)

Qw1 (t) =

∫RA3(z)|∂tw|2 dz +

∫RA4(z)∂tw∂zw dz .

Qw2 (t) = −ε2∫R|∂tw|2 dz + (1− ε2s2)

∫R|∂zw|2 dz + 2ε2s

∫R∂tw∂zw dz +

∫R

(A5(z)− ∂zA4(z)

2

)|w|2 dz .

(5.9)

To determine φ(z) we examinate (5.7)–(5.8). We first require the natural condition ∂zφ(z) ≥ 0.This clearly ensures A3(z) ≥ 1−ε2F ′(0). We examinate the condition A5(z)− 1

2∂zA4(z) ≥ 0 (5.9)in order to fully determine the weight φ(z):

A5(z)− ∂zA4(z)

2= s

(1− ε2F ′(ν)

)∂zφ+ (ε2s2 − 1)

(−∂zzφ+ |∂zφ|2

)+ ε2sF ′′(ν)∂zν − F ′(ν)

− 1

2ε2sF ′′(ν)∂zν + (ε2s2 − 1)∂zzφ

= (ε2s2 − 1) |∂zφ|2 + s(1− ε2F ′(ν)

)∂zφ+

1

2ε2sF ′′(ν)∂zν − F ′(ν) .

18

This is a second-order equation in the variable ∂zφ. Maximization of this quantity is achievedwhen

∂zφ =s(1− ε2F ′(ν)

)2(1− ε2s2)

≥ 0 . (5.10)

We notice that this is equivalent to setting A4(z) = 0 (5.6). Then we obtain

A5(z) =s2(1− ε2F ′(ν)

)24(1− ε2s2)

+1

2ε2sF ′′(ν)∂zν − F ′(ν)

=1

4(1− ε2s2)

(s2(1 + ε2F ′(ν)

)2 − 4F ′(ν))

+1

2ε2sF ′′(ν)∂zν

=1

4 (1− ε2F ′(0))2

(4F ′(0)

(1 + ε2F ′(ν)

)2 − 4F ′(ν)(1 + ε2F ′(0)

)2)+

1

2ε2sF ′′(ν)∂zν

=1

(1− ε2F ′(0))2 (F ′(0)− F ′(ν))

(1− ε4F ′(0)F ′(ν)

)+

1

2ε2sF ′′(ν)∂zν .

We check that A5(z) ≥ 0 since ∀z F ′(ν(z)) ≤ F ′(0), ε2F ′(0) < 1, ∀z F ′′(ν(z)) ≤ 0 and∀z ∂z ν(z) ≤ 0.

We recall that the exponential decay of ν at +∞ is given by the eigenvalue λ > 0, where

λ =s(1− ε2F ′(0))

2(1− ε2s2).

Therefore we can rewrite

∂zφ = λ1− ε2F ′(ν)

1− ε2F ′(0). (5.11)

Remark 17. As far as we are concerned with linear stability, the energies Ew1 and Ew2 containenough information. However proving nonlinear stability requires an additional control of u in L∞

which can be obtained using Eu1 and Eu2 [24].

2- Combination of the energy estimates. We first examinate the energies Eu1 and Eu2 . Weclearly have

Eu2 (t) ≥ − ε4

1− ε2F ′(0)‖∂tu− s∂zu‖22 −

1− ε2F ′(0)

4‖u‖22 +

1− ε2F ′(0)

2‖u‖22

≥ − ε4

1− ε2F ′(0)‖∂tu− s∂zu‖22 +

1− ε2F ′(0)

4‖u‖22

We set

δ =1− ε2F ′(0)

2ε2.

We have on the one hand

Eu1 (t) + δEu2 (t) ≥ 1

2‖∂zu‖22 dz +

∫RA6(z)|u|2 dz ,

where A6(z) is defined as

A6(z) =1

2A1(z) + δ

1− ε2F ′(0)

4.

We have on the other hand,

Qu1 (t) + δQu2 (t) ≥ 1− ε2F ′(0)

2‖∂tu− s∂zu‖22 + δ‖∂zu‖22 +

∫RA7(z)|u|2 dz ,

19

where A7(z) is defined as

A7(z) =s

2∂z(sε2F ′′(ν)∂zν − F ′(ν)

)+ δA2(z) .

We have both limz→−∞A6(z) > 0 and limz→−∞A7(z) > 0. Accordingly there exists α > 0and z0 ∈ R such that

∀z < z0 min(A6(z), A7(z)) > α .

In order to control the zeroth-order terms over (z0,+∞) we shall use the last two energyestimates. First we observe that ∀z > z0 |u(z)| = |e−φ(z)w(z)| ≤ e−φ(z0)|w(z)| since φ is increasing.We set φ(z0) = 0 without loss of generality. This determines completely φ together with thecondition (5.10). We have

Eu1 (t) + δEu2 (t) ≥ 1

2‖∂zu‖22 dz + α

∫z<z0

|u|2 dz −∥∥A6e

−2φ1z>z0∥∥∞

∫z>z0

|w|2 dz ,

Qu1 (t) + δQu2 (t) ≥ 1− ε2F ′(0)

2‖∂tu− s∂zu‖22 + δ‖∂zu‖22 + α

∫z<z0

|u|2 dz −∥∥∥∥A7e

−2φ

A51z>z0

∥∥∥∥∞

∫z>z0

A5(z)|w|2 dz .

Lemma 18. We haveA7e

−2φ

A5∈ L∞(z0,+∞) and

e−φ

A5∈ L∞(z0,+∞).

Proof. The first claim is clearly a consequence of the second claim since A7e−φ ∈ L∞(z0,+∞).

First we have

F ′(0)− F ′(ν) ≥(

inf[0,1]

(−F ′′))ν = αν ,

where α > 0 is the coercivity constant of −F (1.2). As a consequence, A5 ≥(

(1+ε2F ′(0))α1−ε2F ′(0)

)ν.

Second we recall ∂zν + λν ≥ 0 (Lemma 10), so that ∀z > z0, ν(z) ≥ ν(z0)e−λ(z−z0). Finally wehave ∀z, ∂zφ ≥ λ (5.11), thus

∀z > z0, e−φ(z) ≤ e−φ(z0)e−λ(z−z0) ≤ e−φ(z0)

ν(z0)ν(z) ≤ e−φ(z0)

ν(z0)

(1− ε2F ′(0)

(1 + ε2F ′(0))α

)A5(z) .

We now focus on the second series of energy estimates. We clearly have

Ew2 (t) ≥ − ε4

1− ε2F ′(0)‖∂tw‖22 −

1− ε2F ′(0)

4‖w‖22 +

1− ε2F ′(0)

2‖w‖22

≥ − ε4

1− ε2F ′(0)‖∂tw‖22 +

1− ε2F ′(0)

4‖w‖22 ,

and

Qw2 (t) ≥ −ε2‖∂tw‖22 + (1− ε2s2)‖∂zw‖22 −2ε4s2

1− ε2s2‖∂tw‖22 −

1− ε2s2

2‖∂zw‖22 +

∫RA5(z)|w|2 dz

≥ −ε2 1 + ε2s2

1− ε2s2‖∂tw‖22 +

1− ε2s2

2‖∂zw‖22 +

∥∥∥∥e−φA51z>z0

∥∥∥∥−1∞

∫z>z0

e−φ|w|2 dz .

We set

δ′ =1− ε2F ′(0)

2ε2· 1− ε2s2

1 + ε2s2< δ .

We have on the one hand

Ew1 (t) + δ′Ew2 (t) ≥ 1− ε2s2

2‖∂zw‖22 dz +

∫RA8(z)|w|2 dz ,

20

where A8(z) is defined as

A8(z) =1

2A5(z) + δ′

1− ε2F ′(0)

4≥ δ′ 1− ε

2F ′(0)

4.

We have on the other hand,

Qw1 (t) + δ′Qw2 (t) ≥ 1− ε2F ′(0)

2‖∂tw‖22 + δ′

1− ε2s2

2‖∂zw‖22 + δ′

∥∥∥∥e−φA51z>z0

∥∥∥∥−1∞

∫z>z0

h|w|2 dz .

Combining all these estimates, we define E(t) = Ew1 (t) + δ′Ew2 (t) + δ′′ (Eu1 (t) + δEu2 (t)) andQ(t) = Qw1 (t) + δ′Qw2 (t) + δ′′ (Qu1 (t) + δQu2 (t)), where δ′′ > 0 is defined such as the followingcondition holds true

δ′′ < δ′min

(1− ε2F ′(0)

4

∥∥A6e−2φ1z>z0

∥∥−1∞ ,

∥∥∥∥A7e−2φ

A51z>z0

∥∥∥∥−1∞

).

finally obtain our main estimate,

d

dtE(t) +Q(t) ≤ O

(∫R|u||∂tu− s∂zu|2 dz +

∫R|u|3 dz

)+O

(∫Re−φ|w||∂tw|2 dz +

∫Re−φ|w||∂zw|2 dz +

∫Re−φ|w|3 dz

), (5.12)

where

E(t) ≥ O(‖∂zu‖22 +

∫z<z0

|u|2 dz + ‖∂zw‖22 + ‖w‖22),

Q(t) ≥ O(‖∂t − s∂zu‖22 + ‖∂zu‖22 +

∫z<z0

|u|2 dz + ‖∂tw‖22 + ‖∂zw‖22 +

∫z>z0

e−φ|w|2 dz).

3- Control of the nonlinear contributions. Our goal is to control the size of the perturbationu in L∞. For this purpose we use the embeddings of H1(R) into L∞(R) and L4(R):

‖u‖∞ ≤ C‖u‖1/22 ‖∂zu‖1/22 ,

‖u‖4 ≤ C‖u‖3/42 ‖∂zu‖1/42 .

We examinate successively the nonlinear contributions. We recall u = e−φw. First we have∫R|u||∂tu− s∂zu|2 dz ≤ ‖u‖∞‖∂t − s∂zu‖22

≤ O(E(t)1/2Q(t)

),

and similar estimates can be derived for all the contributions in the r.h.s. of (5.12) except for thelast one. Second we have∫

Re−φ|w|3 dz =

∫z<z0

e−φ|w|3 dz +

∫z>z0

e−φ|w|3 dz

≤ ‖u‖L∞(−∞,z0)‖w‖2L2(−∞,z0) + ‖e−φ/2w‖2L4(z0,+∞)‖w‖L2(z0,+∞)

≤ C‖u‖1/2L2(−∞,z0)‖∂zu‖1/2L2(−∞,z0)‖w‖

2L2(−∞,z0)

+ C‖e−φ/2w‖3/2L2(z0,+∞)

∥∥∥∂z (e−φ/2w)∥∥∥1/2L2(z0,+∞)

‖w‖L2(z0,+∞) .

21

We have

‖w‖2L2(−∞,z0) ≤ ‖u‖2L2(−∞,z0) ,

‖e−φ/2w‖2L2(z0,+∞) =

∫z>z0

e−φ|w|2 dz ,∥∥∥∂z (e−φ/2w)∥∥∥2L2(z0,+∞)

≤ 2

∫z>z0

(e−φ|∂zw|2 +

1

4|∂zφ|2e−φ|w|2

)dz ≤ 2

∫z>z0

|∂zw|2 dz + C

∫z>z0

e−φ|w|2 dz .

Consequently we obtain ∫Re−φ|w|3 dz ≤ O

(E(t)1/2Q(t)

).

Finally we getd

dtE(t) +Q(t) ≤ O

(E(t)1/2Q(t)

).

This estimate ensures that the energy is nonincreasing provided that it is initially small enough.Indeed there exists a constant C such that d

dtE(t) +Q(t) ≤ CE(t)1/2Q(t). We set c = C−2/2. Ifinitially E0 ≤ c then the previous differential inequality guarantees that E(t) is decaying and re-mains below the level c. Therefore E(t) is positive decaying, and the dissipation Q(t) is integrable.This concludes the proof of Theorem 15.

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